Radicals for the SAT: why is sqrt[75] = 5sqrt[3] instead of sqrt[5]?

[Illvoices]

Hi, In my DAY practice question there is a question about radicals. Now I need help understanding why the out the most common factor out side of the the sign like so:
√75
5*5*3
√5*5√3
5√3

So why is it the square root of three instead of the square root of five.

 

[Dr.Peterson]

Hi, In my SAT practice question there is a question about radicals. Now I need help understanding why they put the most common factor out side of the sign like so:
√75
√(5*5*3)
√(5*5)√3
5√3

So why is it the square root of three instead of the square root of five.

You should proofread a little. I've fixed several things that didn't make sense, and also added parentheses that are required when you don't typeset.

They split up 75 as 5*5 times 3. Then they split the radical into two, one of 5*5 and the other of 3. The square root of 5*5 = 25 is 5; so that ends up not in the radical, because you have now taken the root -- it doesn't still have to be done. The 3, on the other hand, is still inside a radical, because you haven't taken its root yet.

Here is the work, typeset:
\(\displaystyle \sqrt{75} = \sqrt{5^2\cdot 3} = \sqrt{5^2}\sqrt{3} = 5\sqrt{3}\)